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Questions tagged [fundamental-solution]

Questions on fundamental solutions of an ordinary differential equation.

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Solving the fundamental solution of the origin of $\Delta u=\Delta_x u+\frac{a}{r}u_r+u_{rr}=0$

The discussion starts from introducing a function $u(x,y):\mathbb{R}^n\times\mathbb{R}^{1+a}\to\mathbb{R}$ is radially symmetric, i.e. for $|y|=|y'|=r$, we have $u(x,y)=u(x,y')$. I am working on ...
Christy's user avatar
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Solving the Radially Symmetric Poisson Equation with Exponential Source Term

I want to solve the poisson equation $$ -\Delta u(\mathbf{x})=\rho(\mathbf{x})=\frac{e^{-|\mathbf{x}|^2}}{|\mathbf{x}|^2-1} $$ The problem want me to use the fundamental solution of laplace operator, ...
Gao Minghao's user avatar
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1 answer
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why the particular solution takes this form? [duplicate]

I am trying to find the particular solution of $$ x^{\prime\prime}-3x^{\prime}-4x=5e^{-t} $$ I was always taught that the particular solution takes the same form as the non-homogenious part $5e^{-t}$. ...
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Confusion on definition of fundamental solution for the heat equation

As mentioned on Wikipedia, a fundamental solution for a linear differential operator $L$ is a function (or distribution) $G$ such that $$LG = \delta$$ which by linearity of $L$ gives the following ...
CBBAM's user avatar
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singular matrix and number of eigenvectors

what is the relationship between the a singular matrix and the number of linearly independent eigenvectors? i encountered this question in DE system, and here the number of linearly independent ...
ZOOOOEE's user avatar
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69 views

How is it possible to demonstrate the necessary system to find particular solutions in LDE of order 2

When when want to find particular solutions to a LDE of order 2 like : $$ y'' + ay' + by = f(x) $$ I see in a lot of source that says that after finding two solutions for the homogeneous equations $...
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Question about proof of fundamental solution for Laplacian - Teorem 2.17 Folland, G. Introducction to PDE

I'm reading Introduction to PDE of Folland, G. but I'm stuck in the following theorem: My question is about the $n=2$ case. I tried to do the same argument of $n>2$ but since $N$ and $|\log|x||+1$ ...
matdlara's user avatar
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prove that a solution explodes in finite time

How can I show that the ode $$ \dot x(t) = t^2x^3 \sin(\frac{1}{t^3x}), x(0)=x_0 $$ doesn't have a unique solution on $[-1,1]$ ? It seems $x\equiv 0$ is a solution. Now clearly, for $t,x\neq 0$, $g(t,...
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Constant Coefficient Operators: Fundamental solution of Cauchy-Riemann Operator - Folland, Introduction to PDE

working on the exercises of section F. Constant-Coefficient Operators: Fundamental Solutions but I'm stuck in the following problem: I'm not sure how to use Green's theorem to conclude the second ...
matdlara's user avatar
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Deriving the fundamental solution to heat equation

We are seeking a general solution to the heat equation in the form $$u(t,x)=t^{-\alpha}v\left(\frac{\|x\|^2}{t}\right), x \in \mathbb{R}^n,\alpha>0,t>0$$ for an appropriate function $v:(0,+\...
Weyr124's user avatar
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Fundamental matrix solution remains invertible

Given a system of first-order differential equations that is written as $$ \frac{d}{dt} \vec{x}(t) = A(t) \vec{x}(t) $$ One usually looks at the related matrix differential equation $$ \frac{d}{dt} \...
Bio's user avatar
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Fundamental matrix of $x'= (-1+\frac{3}{2}\cos^2t)x+(1-\frac{3}{2}\sin t\cos t)y,~y'=(-1-\frac{3}{2}\sin t \cos t)x+(-1+\frac{3}{2}\sin^2t) y$

Find the fundamental matrix of the system of odes: $$\left[ \begin{array}{c} x \\ y\\ \end{array} \right]'= \left[ \begin{array}{ccc} -1+\frac{3}{2}\cos^2t & 1-\frac{3}{2}\...
Nikolaos Skout's user avatar
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Show that $\lim_{\tau \to t}\int_\Omega \Phi(x-\xi, t-\tau)u(\xi,\tau)d\xi = u(x,t)$

If $\Omega \subset \mathbb{R}^n$ is a limited domain with a $C^1$ boundary, and $U_T = \{ (x,t) : x \in \Omega, 0 < t< T \}$, I have to show that $\lim_{\tau \to t^-}\int_\Omega \Phi(x-\xi, t-\...
Átila Luna's user avatar
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Verification of the formula of variation of parameters for ODE

Background Consider the non homogeneous linear ODE in $\mathbb{R}^n$: $$\dot{\xi}(t) = A(t)\xi(t) + \nu(t)\; \xi(\sigma_0) = x_0 \label{1}\tag{I}$$ With $A$ a matrix of size $n\times n$ whose entries ...
César VB's user avatar
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Show that $\frac 1{4\pi |x|} e^{-c|x|}$ is a fundamental solution for $-\Delta+c^2$ on $\mathbb R^3$

[Introduction to Partial Differential Equations - Gerald B. Folland, chapter 2, section C, question 6] Show that $$\frac 1{4\pi |r|} e^{-c|r|}$$ is a fundamental solution for $$-\Delta+c^2\qquad (c\in ...
Sayan Dutta's user avatar
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Is this the fundamental solution?

Let $$Tu=\bigg(t\Psi +x \partial_x\bigg)u=0$$ where $\Psi=\partial_{tt}.$ Define a kernel which solves this equation by $$ u(t,x)=\exp\bigg(\frac{t}{\log x} \bigg) $$ for all $t>0$ and $u,x\in(0,1)$...
zeta space's user avatar
2 votes
1 answer
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Finding solution of non-homogeneous ODE when we know the solutions of the homogeneous counterpart

Given ODE is: $$y''+Py'+Qy=0$$ It has two solutions $f(x)$ and $xf(x)$. I need to find the solution to the following non-homogeneous ODE: $$y''+Py'+Qy=f(x)$$ where $f(x)$ on the RHS is the solution of ...
S.S's user avatar
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4 votes
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Exactly one homogeneous differential equation of second order to given fundamental solution

I am working on: Let $\phi_1,\phi_2$, so that $\phi_1(x)\phi_2'(x)-\phi_1'(x)\phi_2(x)\neq 0.$ for all $x\in\mathbb{R}$. Then there exists exactly on homogeneous differential equation of second order $...
john_psl1298's user avatar
3 votes
1 answer
284 views

On the solution of the heat equation using distribution theory

The Green's function $$ \tag{1} \displaystyle K(t,x,y)={\frac {1}{(4\pi t)^{d/2}}}e^{-\|x-y\|^{2}/4t}$$ solves the heat equation $$ {\frac {\partial K}{\partial t}}(t,x,y)=\Delta _{x}K(t,x,y)\ $$ for ...
ric.san's user avatar
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General solution of $x^2 u_{xx} - 2xy u_{xy} + y^2 u_{yy} = e^x$

I am trying to find the general solution of $x^{2} u_{xx} -2xy u_{xy} + y^{2} u_{yy} = e^{x}$. The equation is parabolic as the only root of the characteristic equation is $\lambda = \frac{y}{x}$. ...
PDEsperate's user avatar
5 votes
0 answers
314 views

What is a distinct feature of an ambiguous result

This question comes from my experience in radar signal processing. As I am going more deep into the theory of sampling, statistical signal processing and estimation theory in general, I have a very ...
CfourPiO's user avatar
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Find the fundamental solutions of the following differential operators

Find the fundamental solutions of the following differential operators. Check that they satisfy (outside the singularities) the homogeneous equation in principal variables and the conjugate one in ...
Dmitry's user avatar
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3 votes
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Regarding common zeros of a pair of solutions to an ODE

$\mathbf{Question}:$ Let $y_1$ and $y_2$ be the solutions to $y''x^2+y'+\sin(x)y=0$ which satisfy the boundary conditions $y_1(0)=0$ and $y_1'(1)=1$ and $y_2(0)=1$ and $y_2'(1)=0$ respectively. Then, ...
Subhasis Biswas's user avatar
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What time time do A and B meet?

A starts from X at 9:00 am and reaches Y at 1:00 pm. B starts from Y at 9:00 am and reaches ...
Joypal's user avatar
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How to prove this ODE has solutions that are global?

While studying the existence and uniqueness of the solutions of ODEs, I encountered a problem about local and global solutions, which is as the follows. Consider $I=\mathbb{R}$ and F a $C^1$ vector ...
Chang's user avatar
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Find fundamental matrix in Floquet theory

I have problems finding the fundamental matrix for this exercise of Floquet theory \begin{align*} \dot{x} &= − \sin(2t)x + (\cos(2t) − 1)y\\ \dot{y} &= (\cos(2t) + 1)x + \sin(2t)y \end{align*} ...
Bayesian guy's user avatar
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Let $B\in \mathbb{C}^{n\times m}$, where $n\geq m$, prove that $B^{*}B$ is invertible if and only if $B$ is invertible.

My proof: forward direction: $$(B^{*}B)(B^{*}B)^{-1}=I$$ $$B^{*}B(B^{-1})^{*}B^{-1}=I$$ Only possible when $BB^{-1}=I$. Now, for the backward direction: the same way.
Rust32627's user avatar
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3 votes
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Propagators and PDEs

In quantum field theory one encounters the retarded, advanced and Feynman propagators as certain solutions to a wave equation. Mathematically, these derivations are somewhat magical (typically one ...
Bettina's user avatar
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5 votes
2 answers
141 views

What is the solution to this non-linear second order differential equation?

I'm trying to solve the following non-linear second order differential equation: $$\tag{1} \frac{d\, }{dx} \Bigl( \frac{1}{y^2} \, \frac{dy}{dx} \Bigr) = -\, \frac{2}{y^3}, $$ where $y(x)$ is an ...
Cham's user avatar
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1 answer
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How do you solve for the particular solution of this differential equation?

We were given a differential equation problem and it is somehow mind-boggling that it needed a particular solution. The general solution itself is already very long; let alone a particular solution. ...
Cavern Tooth's user avatar
2 votes
0 answers
130 views

Fundamental solution to $\partial_t = \kappa\partial_x\partial_y f$

The PDE $\partial_t = \kappa\partial_x\partial_y f$, where $\kappa$ is a constant, came up in something I was working on. Boundary conditions are vanishing at infinity. At first, it looks heat ...
eyeballfrog's user avatar
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How to find the fundamental solution of the operator$I-\Delta$?

I want to find a distribution $E$ which satisfies $(I - \Delta) E = \delta$. I tried the usual method of finding the fundamental solution of Laplace operator. Using polar coordinate and calculating $ \...
Tree23's user avatar
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-2 votes
1 answer
132 views

Non-negative fundamental Matrix

Given a system of n homogeneous linear ordinary differential equations $$\dot x = M(t)x(t)$$ where $M(t)$ is a Metzler matrix at any time $t$, prove that the fundamental matrix to this system of ODEs ...
Editor G's user avatar
1 vote
0 answers
127 views

Finding Green function and solution of Dirichlet problem

For unit open ball $D=\{{x}\in\mathbb{R}:\lvert{x}\rvert<1\}$, I need to find Green function $G$ with Dirichlet Problem $$ \Delta u =0\,\, \text{ in }D,\qquad u=g\,\, \text{ on }\partial D $$ Let $\...
Nam Yunjae's user avatar
1 vote
0 answers
56 views

ODE for the inverse of fundamental solution

Let $A:[0,T]\to \mathbb{R}^{n\times n}$ be a continuous function, and let $\Phi$ satisfy $\frac{d}{d t} \Phi_t=A_t \Phi_t$ for all $t\in [0,T]$, and $\Phi_0=I_n$. I hope to prove that the solution to ...
John's user avatar
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1 vote
1 answer
124 views

Solving the integral equation $y(x)=e^{-x^2/2}+\lambda\int^{+\infty}_{-\infty}e^{-i(x-z)}y(z)dz$

Could you please help me to solve the following integral equation? $$y(x)=e^{-x^2/2}+\lambda\int^{+\infty}_{-\infty}e^{-i(x-z)}y(z)dz$$ I tried to turn the exponentiential term into its trigonometric ...
MATAKA's user avatar
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1 vote
0 answers
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Volume potential for Laplace equation in $\mathbb{R}^3$ with $L^p$ source term

It is known that the solution of $$-\Delta u = f \quad \text{in } \mathbb{R}^3$$ can be represented through the volume potential $$ Vf(x) = \int_{\mathbb{R}^3} E(x-y)f(y) \, dy, \quad x \in \mathbb{R}^...
GaC's user avatar
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Fundamental System ODE

Let $\epsilon>0$. Look at the ODE \begin{align*} y'(t)=A(t)y(t), \end{align*} with $A\in C(\mathbb R,\mathbb C^{n\times n})$ and $A(t+\epsilon)=A(t), \forall t\in \mathbb R$. Let $Z\in C^1(\mathbb ...
andy's user avatar
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1 answer
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Proof of invariance of propagators or Green functions from their defining property rather than an explicit expression (Klein-Gordon eq.)

This same question has been asked in the physics section and answered with the usual physics proof, which is unsatisfying to me as I wish to derive the symmetry properties DIRECTLY FROM THOSE OF THE ...
Noix07's user avatar
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1 vote
1 answer
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The asymptotic behavier of the solution of $ f''(r)+\frac{1}{r}f(r)+a^2 f(r)=0 $.

$\;$ Consider the Laplace equation with potential term in $ \mathbb{R}^2 $ as $ \Delta u+a^2u=0 $ where $ a>0 $ is a constant. I am considering the fundamental solution of it, i.e., the solution ...
Luis Yanka Annalisc's user avatar
3 votes
1 answer
545 views

Solving a nonhomogenous system of eqns with one eigenvalue

I have the system: $\left[\begin{array}{@{}c@{}} x' \\ y' \end{array} \right]= \left[\begin{array}{@{}c@{}} 3&2 \\ -2 & -1 \end{array} \right]\left[\begin{array}...
Superunknown's user avatar
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Fundamental Solution / Greens function for Kramers / Langevin equation nowhere to be found

I am looking for a reference to the fundamental solution\Greens function of the two dimensional Kramers PDE $$\partial_t \rho(t,x,v) = -v\frac{\partial}{dx}\rho(t,x,v)+\gamma\frac{\partial}{\partial ...
bus busman's user avatar
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266 views

Intuition for Green's function for the Heat Equation

Here I am interested in the heat equation over the domain $\mathbb{R}_+\times\mathbb{R}^d$. I read this question Green’s Function for the Heat Equation whereby for the heat equation $$\partial_t u= \...
bus busman's user avatar
3 votes
1 answer
130 views

Exact Solution form Fundamental Solution PDE

Let $f$ be a function on $\mathbb{R}_+\times\mathbb{R}^d$. Let $L$ be some differential operator, like $L=\frac{\partial}{\partial t}+\frac{\partial^2}{\partial x^2}$. Consider for some function $g$ ...
bus busman's user avatar
1 vote
0 answers
25 views

Question regarding the fundamental system of an DE

We consider an differential equation $$ d_A: \frac{dW}{dz} = A(z) W$$ with $A(z) \in \mathit{Mat}(n, \mathcal{O}_{\mathbb{C^*}})$ and $W(z)$ a fundamental system of this equation. If I now consider ...
user12345's user avatar
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0 answers
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Exercise with fundamental solution of the heat equation

I have this problem: Let $K(y, t) = \frac{-1}{(4 \pi)^{\frac{n}{2}}} \cdot t^{\frac{-n}{2}} \cdot e^{\frac{- \mid y \mid^2}{4t}}$. If $v \in C^2 (\mathbb{R}^n \times (0, \infty))$, prove that for $t &...
Jean Milt's user avatar
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1 answer
93 views

ODE Solution $y''(x) = y^4(x)$ [closed]

Is it possible to analytically calculate the ODE $y''(x) = y^4(x)$? It's easy to see that $y \equiv 0$, but is there another non-trivial solution to this problem? If not, is there any software to help ...
Santos's user avatar
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488 views

Fundamental solution of heat equation with time dependent diffusion coefficient

Consider the heat equation $$ u_{t} = k(t) \Delta u + f(t,x),~~x \in \mathbb{R}^{n},t>0 $$ $$ u(0,x) = u_{0}(x). $$ If $k(t) = k$, then $$ u(t,x) = \int_{\mathbb{R}^{n}}\phi(x-y,kt)u_{0}(y)dy + \...
Manoj Kumar's user avatar
  • 1,271
1 vote
0 answers
77 views

Structure/Order of explaining distribution-theory

please tell me if this is too off-topic, then I will delete the post. In my master thesis I want to use (and explain) distributions in order to solve some PDE, but since covid still is present it is ...
justabit's user avatar
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1 vote
1 answer
105 views

limit of supremum of Green's function, $\lim_{\epsilon \to 0} (\sup_{z \in \partial B(x,\epsilon)} |G(x,z)|$) for $n \geq 3$

How to determine $\lim_{\epsilon \to 0} (\sup_{z \in \partial B(x,\epsilon)} |G(x,z)|$) for $n \geq 3$? I know that $G(x,z) = \phi (z-x) - w^{x}(z)$. And for $n \geq 3$ the fundamental solution is ...
Laura van Leuven's user avatar

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